In the liked paper why do the equalities in equation 2.7 and 2.11 hold? (the LHS of both the equations is the same and hence the two equations are 2 different ways of writing the full connected functional W)
I guess one reads 2.7 to say that when one is flowing down to the IR from UV one develops only "relevant" (dim <4) operators and one I guess reads 2.11 to mean that one develops only irrelevant (dim >4) operators when one flows up to the UV from IR.
In the linked paper just below equation 2.2 the authors comment that if there is a CFT in the UV then this UV behaviour can change if irrelevant operators are added. why? I would think that (dim>4)/irrelevant operators would come suppressed with positive powers of the cut-off and hence if one pushes the cut-off to infinity then they would vanish and hence the UV is not affected by them. But the authors don't seem to think so...
Regarding your second question: You've got the reasoning backwards (or perhaps a sign wrong in the definitions). Irrelevant deformations of a theory are termed irrelevant because their contributions become less and less important as we zoom out on our field theory. Consequently, they become more and more important as we zoom in. In the UV limit, they are dominant.
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